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Periodic Near Optimal Control
Journal of Mathematical Analysis and Applications 248, 124᎐144 Ž2000. doi:10.1006rjmaa.2000.6874, available online at http:rrwww.idealibrary.com on
Periodic Near Optimal Control G. Grammel Center for Mathematics M6rH4, Technical Uni¨ ersity Munich, Arcisstr. 21, 80290 Munich, Germany E-mail: [email protected] Submitted by Robert E. O’Malley, Jr. Received December 28, 1998
We consider the infinite time optimal control problem of minimizing an average cost functional for nonlinear singularly perturbed systems. In particular, we are interested in the existence of near optimal trajectories which are periodic. For this purpose, we investigate nonlinear control systems with three time scales and construct a limiting system for the slowest motion by an iterated averaging procedure. We furthermore show that global tools, invariant control sets, and local tools, Lie algebra ranks of vector fields, are preserved under singular perturbations. The existence of periodic near optimal trajectories is verified under the condition that the fast subsystem and an unperturbed slow subsystem are controllable and that a standard Lie algebra rank condition for local accessibility holds. 䊚 2000 Academic Press
Key Words: nonlinear control system; periodic control; singular perturbation; near optimality; controllability; averaging.
1. INTRODUCTION We consider the dynamic optimization problem of minimizing an average cost functional 1 T l Ž z Ž t . , y Ž t . , u Ž t . . dt, Ž 1. Ž z 0 , y 0 , u . ¬ lim sup Tª⬁ T 0 where t ¬ z Ž t . s z Ž t, z 0 , y 0 , u. and t ¬ y Ž t . s y Ž t, z 0 , y 0 , u. are the trajectories of the singularly perturbed nonlinear control system
H
˙z Ž t . s f Ž z Ž t . , y Ž t . , u Ž t . . , ⑀˙ y Ž t . s g Ž y Ž t . , uŽ t . . ,
z Ž 0. s z 0 , y Ž 0. s y 0 ,
uŽ t . g ⍀ .
Ž 2.
Here, the singular perturbation parameter ⑀ ) 0 is assumed to be small in order to reflect that y Ž⭈. is much faster than z Ž⭈.. 124 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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A natural aspect arising with this problem is the property of possessing periodic optimal trajectories or at least periodic suboptimal trajectories. The minimization of an average cost functional is a typical infinite time problem, since, for any positive time T, only the costs produced on an infinite time interval w T, ⬁. are relevant in the limit, but not the costs produced on the finite interval w0, T x. On the other hand, existence of periodic Žsub-. optimal controls implies that the minimization problem basically can be reduced to a finite time horizon. By this reason, periodicity of Žsub-. optimal controls only can be expected under special structural properties of the nonlinear control system Ž2.. It is well known, and follows from results in w5x, that the optimization problem Ž1., Ž2. possesses suboptimal periodic trajectories and controls, if the system Ž2. is controllable. When the state space is compact and the system is locally accessible, controllability even is a necessary condition for the existence of periodic suboptimal controls for any average cost functional Žsee, e.g., w15x.. The controllability notion may be weakened, if the dynamics and the functional are somehow adapted. In w5x, it is shown that whenever suboptimal trajectories are contained in a region of complete controllability, a so-called control set, then there exist suboptimal periodic trajectories. Similar structural properties of the control system are required when approximating the average functional by discounted functionals; see, e.g., w20x. In connection with perturbations, the notion suboptimal becomes a little bit confusing. For perturbed systems one is usually interested in controls and trajectories which become closer and closer to optimal ones, as the perturbation tends to zero. Such controls are called near optimal. Moreover, the asymptotic behavior of the optimal value function, as the perturbation tends to zero, becomes an interesting topic, especially for optimization problems on the infinite time horizon. Notice that the averaging techniques for nonlinear control systems or for differential inclusions with two time scales, as developed in w1, 8, 10, 14x, merely result in approximation statements on a bounded time horizon. Theorem 2.5 contains the main result. If the fast subsystem and an averaged slow subsystem, both considered separately, are completely controllable and if a standard Lie algebra rank condition for local accessibility holds, then we have uniform convergence of the optimal value function to a constant function, as ⑀ ª 0, existence of periodic near optimal trajectories. 䢇
䢇
This is a typical singular perturbation result. Instead of assuming controllability of the whole system, we decompose the state space into an unperturbed slow and an unperturbed fast part and require controllability
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only of the decoupled unperturbed subsystems, which are both independent of the perturbation parameter. It is furthermore a nonlinear generalization of well-known results on linear systems. For linear systems, controllability of unperturbed slow and fast subsystems implies already the controllability of the whole singularly perturbed system, at least for small perturbation parameters; see w18x. The proof of Theorem 2.5 consists of two parts. First, we note that the problem has three natural time scales. For this reason we investigate nonlinear control systems with three time scales and construct a limiting system for the slowest motion. In contrast to common averaging approaches to the approximation of control systems with two time scales w10, 14x, we construct a limiting system for the slow motion as a control system and not as a differential inclusion. This method has the advantage that it can be iterated and thus allows an application to systems with multiple time scales. The second part of the proof is of a different nature. Since, for nonlinear singularly perturbed systems, controllability of the fast subsystem and of the averaged slow subsystem does not imply controllability of the whole singularly perturbed system, we furthermore use global tools from dynamical systems theory Žcontrol sets. and local tools from geometric control theory ŽLie algebras of vector fields. and investigate their asymptotic properties, as the singular perturbation parameter tends to zero. It turns out that both tools are somehow preserved under singular perturbations and allow one to obtain enough periodic orbits of the singularly perturbed system. The paper is organized as follows: In Section 2 we give the setting and recall some facts from differential geometry. We furthermore construct an averaged slow subsystem and state the main result. An example from stability theory closes the section. In Section 3 we apply the averaging technique to systems with three natural time scales. Section 4 contains stability statements for control sets and for the Lie algebra rank under singular perturbations. In Section 5 we fit together the results of the previous sections to prove the main result, Theorem 2.5. Section 6 contains the proof of the two scale averaging result introduced in Section 2.
2. PRELIMINARIES AND MAIN RESULT Regularity The state space of the singularly perturbed control system Ž2. is a product M = N, where M respectively N are compact C⬁-manifolds of dimension m respectively n. The control range is a compact metric space
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⍀ and the control functions u: w0, ⬁. ª ⍀ are measurable. The maps l: M = N = ⍀ ª ⺢, f : M = N = ⍀ ª TM, and g: N = ⍀ ª TN are continuous. For every fixed g ⍀ the vector field Ž l Ž⭈, ⭈ , ., f Ž⭈, ⭈ , ., g Ž⭈, .. is Lipschitz continuous on ⺢ = M = N. The regularity of the vector fields guarantees the unique existence of trajectories for any initial points and any control function; see, e.g., w19x. Notation. By d H we denote the Hausdorff metric. By U we denote the set of all measurable control functions u: w0, ⬁. ª ⍀, whereas elements of the control range ⍀ are denoted by . By A⑀ Ž t, Ž z 0 , y 0 .. [ Ž z Ž t, z 0 , y 0 , u., y Ž t, z 0 , y 0 , u.. g M = N: u g U 4 we denote the attainable set of the singularly perturbed system Ž2.. The A¨ eraging Method This order restriction method is based on the idea that, in the limit ⑀ ª 0, the fast state should not be considered explicitly anymore, but only its average influence on the slow dynamics should. This naturally requires some ergodic properties of the fast motion. In its natural time scale s tr⑀ the fast subsystem on N becomes
˙y Ž . s g Ž y Ž . , u Ž . . ,
y Ž 0. s y 0 ,
uŽ . g ⍀ .
Ž 3.
We denote its solutions by y Ž⭈, y 0 , u. and its attainable set by A N Ž , y 0 .. The following controllability assumption on the fast subsystem is essential. Assumption 2.1. The fast subsystem Ž3. is completely controllable on N: There is a finite time max G 0 such that for all y 0 g N the attainable set A N Žw0, max x, y 0 . coincides with N. This controllability assumption guarantees that the fast subsystem has ‘‘enough periodicity’’ and that in the limit all fast initial points have the same rights. It is much weaker than the controllability type condition introduced in w2x, where it is assumed that all sets of velocities g Ž y, . : g ⍀ 4 contain the balls B1Ž0. ; Ty N. In particular, we do not presume the existence of any equilibria of the fast flow on N. The construction of an averaged slow subsystem is similar to the construction introduced in w11, 12x, but without stability assumptions on the fast flow. As in w2, 11, 12x, in some sense, the fast motion will play the role of controls for the unperturbed slow system. For every point y 0 g N we define the subset U 0Ž y 0 . ; U , which consists of 0-periodic control functions u defining 0-periodic trajectories y Ž⭈, y 0 , u.. The subset U 0Ž y 0 . may be empty for certain periods 0 ) 0. But, by the controllability of the fast subsystem, we certainly have a nonvoid union D 0 ) 0 U 0Ž y 0 .. Any periodic fast trajectory of Ž3. defined by
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an initial state y 0 g N and by a corresponding periodic control function u g U 0Ž y 0 . generates an averaged vector field Ž1. on M:
Ž1. Ž z . s
1
0
0
H0 f Ž z, y Ž , y
0
, u . , u Ž . . d .
Let ⍀ Ž1. be the collection of all these vector fields together with its closure in C Ž M; TM .. Then ⍀ Ž1. can be considered as the control range of an averaged slow subsystem on M,
˙z Ž t . s f Ž1. Ž z Ž t . , uŽ1. Ž t . . ,
z Ž 0. s z 0 ,
uŽ1. Ž t . g ⍀ Ž1. ,
Ž 4.
where the right-hand side is defined by f Ž1. Ž z, Ž1. . [ Ž1. Ž z .. We denote its solutions by z 0 Ž⭈, z 0 , uŽ1. .. In order to get a richer controllability structure of the whole singularly perturbed system, we add an analogous controllability condition on the averaged slow subsystem Ž4.. We denote the attainable set for Ž4. by A M Ž t, z 0 .. Assumption 2.2. The averaged slow subsystem Ž4. is completely controllable on M: There is a finite time t max G 0 such that for all z 0 g M the attainable set A M Žw0, t max x, z 0 . coincides with M. Nevertheless, there are simple two-dimensional examples on the torus which show that this ‘‘composite controllability’’ as introduced with Assumptions 2.1 and 2.2 does not guarantee the existence of periodic trajectories of the singularly perturbed system. Thus, the standard Lie algebra rank condition on nonlinear control systems seems to be vital for the ‘‘synchronization’’ of the two subsystems. It requires higher regularity of the vector fields Ž f, g .. Assumption 2.3. For any fixed g ⍀ the vector field Ž f Ž⭈, ⭈ , ., g Ž⭈, .. is C⬁ on M = N. The Lie algebra generated by the vector fields f Ž⭈, ⭈, ., g Ž⭈, .. has full rank m q n at some Ž z 0 , y 0 . g M = N. DEFINITION 2.4. Consider the optimization problem Ž1., Ž2.. We define the corresponding optimal value function on M = N as V⑀ Ž z 0 , y 0 . [ inf
ug U
ž
lim sup Tª⬁
1
T lŽ zŽt, z H T 0
0
, y 0 , u Ž ⭈. . ,
y Ž t , z 0 , y 0 , u Ž ⭈ . . , u Ž t . . dt .
/
Now we are in a position to state the main result of the paper.
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THEOREM 2.5. ŽI. Suppose Assumptions 2.1 and 2.2 hold. Then there is a continuous function
: w 0, ⬁ . ª w 0, ⬁ .
with Ž 0 . s 0,
and a real number V0 g ⺢ such that for all Ž z 0 , y 0 . g M = N the estimate V⑀ Ž z 0 , y 0 . y V0 F Ž ⑀ . is ¨ alid. ŽII. If additionally Assumption 2.3 holds, then, for ⑀ ) 0 small enough, there exists a set I⑀ ; M = N with d H Ž I⑀ , M = N . F Ž ⑀ . , such that for all initial states Ž z 0 , y 0 . g I⑀ there exists a period t 0 ) 0 and a t 0-periodic control function u⑀ producing a t 0-periodic trajectory Ž z⑀ Ž⭈., y⑀ Ž⭈.. of the system Ž2. with 1 t0
t0
H0
l Ž z⑀ Ž t . , y⑀ Ž t . , u⑀ Ž t . . dt y V0 F Ž ⑀ . .
What follows is a short discussion of the conditions of this result. The controllability condition on the fast subsystem, Assumption 2.1, is very natural, if one wants to achieve that the averaged slow subsystem does not depend on fast initial states. We furthermore assume that the fast flow is decoupled, in the sense that the vector field g does not depend on the slow variable z. This restriction is necessary in order to avoid a significant loss of regularity of an averaged system. Moreover, our averaging procedure would not work this way, if we allow a fully coupled system, since it is not clear how the periodic trajectoriesrcontrols of the unperturbed fast flow depend on the slow state. The controllability condition on the averaged slow subsystem, Assumption 2.2, may be difficult to verify for practical problems. Nevertheless, it is a condition for an ⑀-independent system on a lower dimensional space M and for this reason it is easier to check than controllability of the full system on M = N for e¨ ery ⑀ ) 0. Assumption 2.3 is reduced to the minimum, since we assume full rank of the Lie algebra in only one point of the state space. We give a low-dimensional example from stability theory. EXAMPLE 2.6. We consider a singularly perturbed controlled oscillator defined by 0
˙s Ž t . s y1 q y Ž t . u Ž t . 1
1 s t , 0 Ž .
⑀˙ yŽ t. s
0 y1
1 yŽ t. , 0
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where sŽ t . s Ž s1Ž t ., s2 Ž t .. g ⺢ 2 and y Ž t . s Ž y 1Ž t ., y 2 Ž t .. g ⺢ 2 with 5 y Ž t .5 ' 1. Let the control range ⍀ be an interval w min , max x. We investigate the exponential stability of the slow subsystem via the exponential growth rates
⑀ Ž s 0 , y 0 , u . s lim sup Tª⬁
1 T
log Ž s Ž T , s 0 , y 0 , u .
..
The projection onto the sphere M s S 1 defines a slow subsystem for z Ž t . s sŽ t .r5 sŽ t .5: 0
1 z t 0 Ž .
˙z Ž t . s y1 q y Ž t . u Ž t . 1
0 y zŽ t. , y1 q y 1 Ž t . u Ž t .
¦
1 z t z t . 0 Ž . Ž .
;
Moreover, in these coordinates, the exponential growth rates can be written as an average functional
⑀ Ž z 0 , y 0 , u . s lim sup Tª⬁
1
T
H T 0
¦
zŽ t. ,
0 y1 q y 1 Ž t . u Ž t .
1 z t dt. 0 Ž .
;
It is obvious that the fast subsystem on N s S 1 is periodic. Hence, Assumption 2.1 is satisfied and we can construct an averaged slow subsystem. The averaged vector fields are of the form
Ž1. Ž z . s
1
2 k
H 2 k 0
ž
0 y1 q y 1 Ž t . u Ž t .
¦
y z,
1 z 0
0 y1 q y 1 Ž t . u Ž t .
1 z z dt, 0
;/
where k g ⺞ and u g U . In particular, for k s 1 and uŽ t . ' min , we obtain an averaged vector field
Ž1. Ž z . s
0 y1
1 0 z y z, 0 y1
¦
1 0 z zs 0 y1
;
1 z, 0
which produces periodic trajectories on S 1 for the averaged slow subsystem. Hence, Assumption 2.2 is satisfied and we can apply Theorem 2.5, part ŽI., and obtain that the minimal exponential growth rates converge, as ⑀ ª 0.
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As for Assumption 2.3, we see immediately that the vectors
and
0 y1 q y 1 min
1 0 0 z y z, y1 q y 1 min
¦
0 y1
0 y1 q y 1 max
1 y 0
1 0 0 z y z, y1 q y 1 max
¦
0 y1
1 y 0
1 0 z z
;
0 ; 0
1 0 z z
are linearly independent in Tz M = Ty N, if min - max and if z1 y 1 / 0. Thus, Assumption 2.3 is satisfied and we conclude that the minimal exponential growth rates can be Žnearly. realized by periodic controls.
3. ORDER REDUCTION VIA ITERATED AVERAGING The averaged slow subsystem serves us as a nominal system for the slow motion in the following sense: Let S⑀ Ž z 0 , y 0 . ; C Žw0, T x, M . be the set of all slow trajectories z Ž⭈. of the singularly perturbed control system Ž2. and let S0 Ž z 0 . ; C Žw0, T x, M . be the set of all trajectories of the averaged slow subsystem Ž4., both obtained for times t g w0, T x. Then we have: LEMMA 3.1. Suppose Assumption 2.1 holds. The control range ⍀ Ž1. of the a¨ eraged slow subsystem Ž4. is a compact metric space, the map f Ž1. : M = ⍀ Ž1. ª TM is continuous, and for e¨ ery fixed Ž1. g ⍀ Ž1. the ¨ ector field f Ž1. Ž⭈, Ž1. . is Lipschitz continuous on M. Furthermore there is a constant C G 0 such that for all initial ¨ alues Ž z 0 , y 0 . g M = N, perturbation parameters ⑀ ) 0, and times T G 1 the following estimate holds: d H Ž S⑀ Ž z 0 , y 0 . , S0 Ž z 0 . . F e C T⑀ 1r3 . The proof is contained in the last section. Lemma 3.1 is of interest in its own right for two reasons: It provides an averaging procedure which preserves the system’s structure and it gives explicit approximation rates of order O Ž ⑀ 1r3 . for the uniform convergence of the slow trajectories on finite time intervals. It is remarkable that the rate of convergence coincides with the one obtained in w9x. There, a singularly perturbed ODE with decoupled fast
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flow is considered and it is shown that the average rate of convergence of the slow trajectories is of order O Ž ⑀ 1r3 ., where the average is taken over the fast initial states in N with respect to a canonical measure. The construction of the averaged slow subsystem is different than the usual averaging techniques, since it preserves the regularity of the system and thus a re-iteration of the procedure becomes possible. In contrast, the usual averaging techniques, as in w10, 14x, produce an averaged slow subsystem in the form of a differential inclusion. Notice that the dynamic optimization problem Ž1., Ž2. possesses three natural time scales. Since ⑀ ) 0 is assumed to be small, y Ž⭈. moves faster than z Ž⭈.. On the other hand the integrand of the cost functional becomes smaller and smaller since we are interested in the situation T ª ⬁. For this reason we study the approximation properties of an augmented control system, which is set up by introducing a new trajectory cŽ⭈. and a perturbation parameter ␦ ) 0:
˙c Ž t . s ␦ e Ž c Ž t . , z Ž t . , y Ž t . , u Ž t . . ,
c Ž 0. s c 0 ,
˙z Ž t . s f Ž z Ž t . , y Ž t . , u Ž t . . ,
z Ž 0. s z 0 ,
⑀˙ y Ž t . s g Ž y Ž t . , uŽ t . . ,
y Ž 0. s y , 0
Ž 5. uŽ t . g ⍀ .
We assume that the state space of the additional cŽ⭈.-trajectory, K, is the Euclidean space ⺢ k or a compact C⬁-manifold of dimension k. The map e: K = M = N = ⍀ ª TK is continuous and for any fixed g ⍀ the map eŽ⭈, ⭈ , ⭈ , . is Lipschitz continuous on K = M = N. The connection with the optimization problem becomes clear by setting K s ⺢, eŽ c, z, y, . [ l Ž z, y, ., c 0 [ 0, and ␦ [ 1rT and by investigating the augmented system on the time interval t g w0, 1r␦ x. We work on this more general augmented system because all approximation statements that follow in this section are valid in this generality. The optimization problem suggests that one first considers the limit ␦ ª 0 for a fixed perturbation parameter ⑀ ) 0 and then investigates the behavior of suboptimal trajectories for this problem, as ⑀ ª 0. But from an approximation point of view it seems to be more natural to first eliminate the fastest motion y Ž⭈. and then to investigate the optimization problem for a limit system describing the situation ⑀ s 0. The latter procedure of eliminating the fastest motion furthermore has the advantage that it can be iterated. So a limiting system describing the limit Ž ⑀ , ␦ . s Ž0, 0. for the slowest motion cŽ⭈. canonically can be constructed. We start the order reduction procedure by applying Lemma 3.1 and considering Ž cŽ⭈., z Ž⭈.. as the slow and y Ž⭈. as the fast trajectories of the augmented system Ž5.. The construction of an averaged system for the Ž cŽ⭈., z Ž⭈.. trajectories representing the situation ⑀ s 0 is the same as
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before: Any periodic fast trajectory of Ž3. defined by an initial value y 0 g N and by a corresponding periodic control function u g U 0Ž y 0 . generates an averaged vector field Ž1. on K = M:
Ž1. Ž c, z . s
1
0
0
H0
ž
e Ž c, z, y Ž , y 0 , u . , u Ž . . f Ž z, y Ž , y 0 , u . , u Ž . .
/
d .
Let ⍀ Ž1. be the collection of all these vector fields together with its closure in C Ž K = M; TK = TM .. Then ⍀ Ž1. can be considered as the control range of an averaged control system on K = M:
˙c Ž t . s ␦ e Ž1. Ž c Ž t . , z Ž t . , uŽ1. Ž t . . ,
c Ž 0. s c 0 ,
˙z Ž t . s f Ž1. Ž z Ž t . , uŽ1. Ž t . . ,
z Ž 0. s z 0 ,
uŽ1. Ž t . g ⍀ Ž1. .
Ž 6.
We use the same notation f Ž1., uŽ1., ⍀ Ž1. for the augmented system Ž6. as for Ž4., since the control action on the Ždecoupled. system for z Ž⭈. is the same. By Assumption 2.2, for every point z 0 g M we can define the subset Ž1. Ž 0 . Ut 0 z ; U Ž1., which consists of t 0-periodic control functions uŽ1. defining t 0-periodic trajectories z Ž⭈, z 0 , uŽ1. . of Ž6.. Any t 0-periodic z Ž⭈, z 0 , uŽ1. . trajectory of Ž6. generates an averaged vector field Ž2. on K
Ž2. Ž c . s
1 t0
t 0 Ž1.
H0
e
Ž c, z Ž t , z 0 , uŽ1. . , uŽ1. Ž t . . dt.
Again, considering the closure of these vector field in C Ž K, TK . as a new control range ⍀ Ž2., we get a limit system for the c-motion on K:
˙c Ž t . s ␦ e Ž2. Ž c Ž t . , uŽ2. Ž t . . ,
c Ž 0. s c 0 ,
uŽ2. Ž t . g ⍀ Ž2. ,
Ž 7.
To describe the connection between this limit system Ž7. on K and the original augmented control system Ž5. on K = M = N, let SŽ ⑀ , ␦ .Ž c 0 , z 0 , y 0 . ; C Žw0, 1r␦ x; K ., respectively SŽ0, 0.Ž c 0 . ; C Žw0, 1r␦ x; K ., be the set of solutions of the augmented system Ž5., projected onto the slowest motion cŽ⭈., respectively of the limit system Ž7.. PROPOSITION 3.2. Suppose Assumptions 2.1 and 2.2 hold. The control range ⍀ Ž2. of the limit system Ž7. is a compact metric space, the map e Ž2. : K = ⍀ Ž2. ª TK is continuous, and for e¨ ery fixed Ž2. g ⍀ Ž2. the ¨ ector field e Ž2. Ž⭈, Ž2. . is Lipschitz continuous on K. Furthermore there are continuous functions
 , ␦ : w 0, ⬁ . ª w 0, ⬁ .
with  Ž 0 . s ␦ Ž 0 . s 0
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such that for all initial ¨ alues Ž c 0 , z 0 , y 0 . g K = M = N and perturbation parameters ⑀ ) 0 the following estimate holds: d H Ž SŽ ⑀ , ␦ Ž ⑀ .. Ž c 0 , z 0 , y 0 . , SŽ0 , 0. Ž c 0 . . F  Ž ⑀ . . Proof. The regularity of e Ž2. and the approximation statement followed by an iterated application of Lemma 3.1: Let SŽ0, ␦ .Ž c 0 , z 0 . ; C Žw0, 1r␦ x, K . be the set of solutions of Ž6., projected onto the slowest motion of cŽ⭈.. Lemma 3.1 applied to Ž5. and Ž6. gives the estimate d H Ž SŽ ⑀ , ␦ . Ž c 0 , z 0 , y 0 . , SŽ0 , ␦ . Ž c 0 , z 0 . . F e C r ␦⑀ 1r3 . Lemma 3.1 applies to Ž6. and Ž7. gives the estimate d H Ž SŽ0 , ␦ . Ž c 0 , z 0 . , SŽ0 , 0. Ž c 0 . . F e C␦ 1r3 . Thus we have for all Ž c 0 , z 0 , y 0 . g K = M = N d H Ž SŽ ⑀ , ␦ . Ž c 0 , z 0 , y 0 . , SŽ0 , 0. Ž c 0 . . F e C r ␦⑀ 1r3 q e C␦ 1r3 , and the claim follows, e.g., setting ␦ Ž ⑀ . [ yCrlogŽ ⑀ 1r6 . for ⑀ g Ž0, 1r2x.
4. CONTROL SETS AND LOCAL ACCESSIBILITY In this section we use the ‘‘composite controllability,’’ introduced with Assumptions 2.1 and 2.2, to get global approximate controllability properties of the singularly perturbed system Ž2. for sufficiently small perturbation parameters ⑀ ) 0. We furthermore show that the standard Lie algebra rank condition, Assumption 2.3, is robust under singular perturbations. Since the Lie algebraic condition is related to local controllability properties, we get a complete picture of the controllability structure of the singularly perturbed system Ž2.. In the sequel we study forward invariant subsets of the state space, in which one can steer at least approximately from any point to any other. These subsets are introduced in w17x, where they are called in¨ ariant control sets. For more details and perturbation results of control sets consult, e.g., w6, 13x. DEFINITION 4.1. We call a subset C ; M = N an invariant control set of Ž2. if clos C s clos AŽw0, ⬁., Ž z 0 , y 0 .. for all Ž z 0 , y 0 . g C and if C is maximal Žw.r.t. set inclusion. with this property. We show that the invariant control sets of the singularly perturbed system Ž2. are becoming denser in M = N, as the perturbation becomes smaller.
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135
LEMMA 4.2. Suppose Assumptions 2.1 and 2.2 hold. For all ⑀ ) 0 there is at least one closed in¨ ariant control set C⑀ ; M = N of the singularly perturbed control system Ž2.. There is a continuous function : w0, ⬁. ª w0, ⬁. with Ž0. s 0 such that for any family of in¨ ariant control sets C⑀ ; M = N 4⑀ ) 0 of the singularly perturbed control system Ž2. the following estimate holds: d H Ž C⑀ , M = N . F Ž ⑀ . . Proof. By compactness of M = N there exists at least one closed invariant control set C⑀ ; M = N for any ⑀ ) 0. This can be proved as follows: We assume that the right-hand side of the singularly perturbed system Ž2. is convex in TŽ z, y.Ž M = N . for all Ž z, y . g M = N. Then the attainable sets define a continuous semi-flow on the system of compact subsets of M = N. By Zorn’s lemma there is a compact subset C⑀ which is invariant under the attainable set semi-flow and minimal with this property; i.e., no other compact subset of C⑀ is invariant. Obviously this minimal invariant set C⑀ is a closed invariant control set of the singularly perturbed system. If the right-hand sides of the system are not convex, we still can work with the corresponding convex valued differential inclusion, and the obtained invariant control set also is a control set for the nonconvex system by well-known relaxation theorems. We take a family of closed invariant control sets C⑀ ; M = N 4⑀ ) 0 . Let Ž z⑀0 , y⑀0 . g C⑀ . By Lemma 3.1 and by Assumption 2.2 the projection ⌸ M A⑀ Žw0, 1 q t max x, Ž z⑀0 , y⑀0 .. onto M fulfills the estimate d H Ž ⌸ M A⑀ Ž w 0, 1 q tmax x , Ž z⑀0 , y⑀0 . . , M . F e CŽ1qt ma x .⑀ 1r3 . By Assumption 2.2 the projection ⌸ N A⑀ Žw0, max ⑀ x, Ž z⑀0 , y⑀0 .. onto N coincides with N, whereas obviously d H Ž ⌸ M A⑀ Ž w 0, max ⑀ x , Ž z⑀0 , y⑀0 . . , z⑀0 . F P˜⑀max , where P˜ G 0 is a uniform upper bound for 5 f 5. Since A⑀ Žw0, 1 q t max q max x, Ž z⑀0 , y⑀0 .. ; C⑀ we conclude that d H Ž C⑀ , M = N . F e C Ž1qt ma x .⑀ 1r3 q P˜⑀max , and the proof is finished. DEFINITION 4.3. The singularly perturbed system Ž2. is locally accessible in Ž z 0 , y 0 . g M = N, if for all t 0 ) 0 int A⑀ Ž w 0, t 0 x , Ž z 0 , y 0 . . / ⭋
and
int A⑀ Ž w yt 0 , 0 x , Ž z 0 , y 0 . . / ⭋.
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G. GRAMMEL
For nonlinear control systems, which are produced by C⬁-vector fields, local accessibility can be verified via the calculation of the Lie products and of the corresponding Lie algebra. If the Lie algebra L , generated by all admissible vector fields, has full rank, then the control system is locally accessible Žcompare w16, 19x.. Now we investigate the dependence of the Lie algebra rank condition for local accessibility on the singular perturbation parameter. Since the limit vector fields Ž0, g Ž⭈, .. s lim ⑀ ª 0 Ž ⑀ f Ž⭈, ⭈ , ., g Ž⭈, .. are degenerated on M = N, it is not obvious whether this condition is fulfilled for sufficiently small singular perturbation parameters. For more convenience we consider the vector fields in the natural scale of the fast motion. LEMMA 4.4. Suppose Assumption 2.3 holds. Then there is an ⑀ 0 ) 0 and a radius r ) 0 such that for all ⑀ g Ž0, ⑀ 0 x and all Ž z, y . g Br Ž z 0 , y 0 . [ Ž z, y . g M = N : dŽŽ z, y ., Ž z 0 , y 0 .. F r 4 the Lie algebra L⑀ , generated by the ¨ ector fields Ž ⑀ f Ž⭈, ⭈ , ., g Ž⭈, .., has full rank m q n at Ž z, y .. Proof. By continuity of the vector fields, there is a radius r ) 0 such that the Lie algebra L1 , generated by the vector fields Ž f Ž⭈, ⭈ , ., g Ž⭈, .., has full rank m q n at Ž z, y ., for all Ž z, y . g Br Ž z 0 , y 0 .. We fix a pair Ž z, y . g Br Ž z 0 , y 0 .. For ⑀ s 1 the linear space in TŽ z, y .Ž M = N . generated by the Lie algebra L⑀ coincides with the full tangent space TŽ z, y.Ž M = N .. In other words, for ⑀ s 1, there are m q n vectors ¨ 1Ž z, y, ⑀ ., . . . , ¨ mq nŽ z, y, ⑀ .4 in the Ž m q n.-dimensional tangent space TŽ z, y .Ž M = N ., which are defined by vector fields ¨ 1Ž⭈, ⭈ , ⑀ ., . . . , ¨ mqnŽ⭈, ⭈ , ⑀ .4 ; L⑀ on M = N such that det Ž ¨ 1 Ž z, y, ⑀ . , . . . , ¨ mqn Ž z, y, ⑀ . . / 0.
Ž 8.
We observe that the determinant detŽ ¨ 1Ž z, y, ⑀ ., . . . , ¨ mqnŽ z, y, ⑀ .. can be written as a real polynomial in ⑀ . We conclude that there is an ⑀Ž z, y . ) 0 such that Ž8. holds for all ⑀ g Ž0, ⑀Ž z, y . x. The claim follows by compactness of Br Ž z 0 , y 0 .. Lemmas 4.2 and 4.4 show that invariant control sets and Lie algebraic conditions on the vector fields are in some sense robust under singular perturbations. Together they ensure that invariant control sets have nonvoid inferior for sufficiently small perturbation parameters.
5. PROOF OF THEOREM 2.5 Theorem 2.5 consists of two parts. In the proof of the first part, ŽI., we only use Assumptions 2.1 and 2.2 to show via a three scale analysis Žsee Proposition 3.2 above. the existence of Žnot necessarily periodic. near optimal trajectories for any initial state Ž z 0 , y 0 . of Ž2..
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In the proof of the second part, ŽII., we make use of the controllability type results obtained in the previous section in order to show that for sets of initial pairs Ž z 0 , y 0 ., which become denser in M = N, the near optimal trajectories can be closed to periodic near optimal trajectories. PROOF OF ŽI.. We first show that to every initial point Ž z 0 , y 0 . g M = N corresponds a family of near optimal, not necessarily periodic, control functions. Let Ž z 0 , y 0 . g M = N and ⑀ ) 0. We set T [ T⑀ [ 1r␦ Ž ⑀ ., where ␦ Ž ⑀ . ) 0 is given in Proposition 3.2. Then we get a three time scale control system Ž5. by setting e Ž c, z, y, . [ l Ž z, y, . ,
K [ ⺢,
␦[␦Ž⑀.,
c 0 s 0.
By Proposition 3.2 we can build up a limit system Ž7. for the slowest motion cŽ⭈.. Since the vector fields eŽ⭈, ⭈ , . do not depend on c g K, Ž7. equivalently can be written as
˙c Ž t . s uŽ2. Ž t . ,
c Ž 0 . s 0,
uŽ2. Ž t . g ⍀ Ž2. .
Now let T [ kT⑀ for a natural number k. Then for control functions u: w0, T x ª ⍀ the slowest motion of the augmented system Ž5. corresponding to the initial value c 0 [ 0 fulfills cŽ T . s s
1 T 1 k
T
H0
k
Ý is1
l Ž z Ž t . , y Ž t . , u Ž t . . dt 1
iT⑀
H l Ž z Ž t . , y Ž t . , u Ž t . . dt. T Ž iy1.T ⑀
⑀
According to Proposition 3.2 for all control functions u: wŽ i y 1.T⑀ , iT⑀ x ª ⍀ the following holds: 1
iT⑀
H l Ž z Ž t . , y Ž t . , u Ž t . . dt G min T Ž iy1.T ⑀
Ž2.
g ⍀ Ž2. 4 y  Ž ⑀ . .
⑀
Again by Proposition 3.2 the control function u: wŽ i y 1.T⑀ , iT⑀ x ª ⍀ can be chosen such that 1
iT⑀
H l Ž z Ž t . , y Ž t . , u Ž t . . dt F min T Ž iy1.T ⑀
Ž2.
g ⍀ Ž2. 4 q  Ž ⑀ . .
⑀
Then we obtain c Ž T . y min Ž2. g ⍀ Ž2. 4 F  Ž ⑀ . .
Ž 9.
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G. GRAMMEL
Since k is arbitrary, we conclude that for all Ž z 0 , y 0 . and ⑀ ) 0 the following holds: V⑀ Ž z 0 , y 0 . y min Ž2. g ⍀ Ž2. 4 F  Ž ⑀ . . Thus, every initial value Ž z 0 , y 0 . g M = N corresponds to a family of near optimal control functions. Furthermore the optimal values V⑀ Ž z 0 , y 0 . uniformly converge to V0 [ min Ž2. g ⍀ Ž2.4 . Proof of ŽII.. We now show that for initial values Ž z 0 , y 0 . g M = N, which lie in the interior of invariant control sets, the corresponding near optimal controls can be closed to periodic near optimal controls. We define I⑀ [ int C⑀ for invariant control sets C⑀ ; M = N of the singularly perturbed control system Ž2.. By Lemma 4.4 there is an ⑀ 0 ) 0, a radius r ) 0, and a point Ž z 0 , y 0 . g M = N such that the singularly perturbed control system is locally accessible in Br Ž z 0 , y 0 . for ⑀ g Ž0, ⑀ 0 x. On the other hand, for Ž ⑀ . - r we have C⑀ l Br Ž z 0 , y 0 . / ⭋. By the forward invariance of the invariant control set C⑀ we conclude I⑀ / ⭋ for ⑀ g Ž0, ⑀ 0 x and that furthermore C⑀ s clos I⑀ . By Lemma 4.2 we have d H Ž I⑀ , M = N . F Ž ⑀ .. We take Ž z 0 , y 0 . g I⑀ and ⑀ g Ž0, ⑀ 0 x. There is a time t⑀ Ž z 0 , y 0 . G 0 such that all points in C⑀ can be steered to Ž z 0 , y 0 . in a positive time smaller than t⑀ Ž z 0 , y 0 .. This follows easily from the local accessibility and the compactness of C⑀ : By local accessibility there is an open ball B ; I⑀ g Ž z, y . g M = N: Ž z 0 , y 0 . g A⑀ Žw0, 1x, Ž z, y ..4 and, by compactness of C⑀ , all points of C⑀ can be steered into this ball in uniformly bounded time. Thus any trajectory starting in Ž z 0 , y 0 . g I⑀ , after arbitrary long time, can be steered back to Ž z 0 , y 0 . g I⑀ in uniformly bounded time. We take a natural number k large enough that 2 ⍀ Ž2. t⑀ Ž z 0 , y 0 . kT⑀
F Ž⑀.
and set T [ kT⑀ . A periodic near optimal control function is constructed by setting u⑀ Ž t . [ u Ž t .
for t g w 0, T x ,
where u: w0, T x ª ⍀ is chosen according to Ž9., and by extending it on w T, T q t⑀ Ž z 0 , y 0 .x in such a way that for a time t 0 g w T, T q t⑀ Ž z 0 , y 0 .x the trajectory of the singularly perturbed control system Ž2. is back in Ž z 0 , y 0 . g I⑀ ; C⑀ . Then a straightforward calculation shows that 1 t0
t0
H0
l Ž z Ž t . , y Ž t . , u⑀ Ž t . . dt F V⑀ Ž z 0 , y 0 . q 2  Ž ⑀ . .
Finally, we define ⑀ [ max 3  Ž ⑀ ., ⑀ 4 and the proof is finished.
PERIODIC NEAR OPTIMAL CONTROL
139
6. PROOF OF LEMMA 3.1 We recall some facts from differential geometry which will be needed in the following proof. We assume that the manifold M is provided with a fixed Riemannian metric which naturally defines a metric dŽ⭈, ⭈ . on the tangent bundle TM Žcompare w7, pp. 78᎐80x.. Since M is compact we can manage with a finite family of local parametrizations i : ⺢ m > Ui ª M to cover the whole manifold. Note that a local parametrization i : ⺢ m > Ui ª M of the manifold M automatically extends to a local parametrization ⌽i : Ui = ⺢ m ª TM of the tangent bundle TM Žcompare w4, pp. 335, 336x.. Furthermore, again by compactness of M, there is a positive number ␣ ) 0 such that for all z g M there is a local parametrization whose image on M contains the closed ball B␣ Ž z . ; M. Finally we remark that the distances on the tangent bundle locally can be estimated by distances in Ui = ⺢ m and vice versa Žsee w3, pp. 126, 126x.. To be more precise: There is a constant ␥ ) 0 such that for all z g M, all z k g B␣ Ž z . ; M, and all ¨ k g Tz k M Ž k s 1, 2. the following holds: y1 ␥ d Ž Ž z1 , ¨ 1 . , Ž z1 , ¨ 2 . . F ⌽y1 i Ž z1 , ¨ 1 . y ⌽i Ž z 2 , ¨ 2 .
F
1
␥
d Ž Ž z1 , ¨ 1 . , Ž z1 , ¨ 2 . . .
Ž 10 .
By L G 0 we denote the Lipschitz constant of the functions f Ž⭈, y, ., considered in local coordinates. According ␥ 2 L G 0 is a Lipschitz constant of the functions f Ž⭈, y, ., considered on the manifold M. By P G 0 we denote the upper bound of the functions 5 f Ž⭈, y, .5, considered in local coordinates. Accordingly ␥ P G 0 is an upper bound of the functions dŽŽ⭈, 0., Ž⭈, f Ž⭈, y, ...: M ª TM. In the proofs we do not change notations for points and tangent vectors when they are considered in local coordinates. We also assume that the manifold N is provided with a fixed Riemmanian metric. Proof of Lemma 3.1. Ži. Obviously, the averaged vector fields Ž1. on M, generated by periodic controls u g U 0Ž y 0 ., are globally Lipschitz continuous on M, since we have in local coordinates for z1 , z 2 g y1 B␣ Ž z 0 .
Ž1. Ž z1 . y Ž1. Ž z 2 . F
1
0
0
H0
f Ž z1 , y Ž , y 0 , u . , u Ž . . y f Ž z 2 , y Ž , y 0 , u . , u Ž . . d
F L 5 z1 y z 2 5
140
G. GRAMMEL
are accordingly with respect to the Riemannian metric d Ž Ž1. Ž z1 . , Ž1. Ž z 2 . . F ␥ 2 Ld Ž z1 , z 2 . . Thus, by the Arzela Ascoli theorem, the closure of these vector fields in C Ž M, TM . is compact and f Ž1. of the required regularity. Žii. The right-hand sides of the averaged slow subsystem Ž4.,
D f Ž1. Ž z, Ž1. . 4 ,
FŽ z. [
g⍀ Ž1. Ž1.
can be represented as limits of periodic a¨ erages Fper Ž z, y 0 , S . [
D
D
0g Ž0, S . ug U 0 Ž y 0 .
½
1
0
0
H0 f Ž z, y Ž , y
0
5
, u . , u Ž . . dt .
By construction of the averaged slow subsystem we have F Ž z . s clos D Fper Ž z, y 0 , S . . S)0
Note that by controllability of the fast subsystem, the right-hand sides F Ž z . do not depend on the initial value y 0 g N and furthermore necessarily are convex, since with increasing S G S0 ) 0 more and more convex combinations of Fper Ž z, y 0 , S0 . are included in Fper Ž z, y 0 , S .. In order to compare the right-hand sides F Ž z . and the periodic averages Fper Ž z, y 0 , S ., we compare both of them with the finite time a¨ erages F Ž z, y 0 , S . [ clos
D
ug U
½
1
S f Ž z, y Ž , y H S 0
0
5
, u . , u Ž . . d .
By controllability of the fast subsystem there is a constant C1 G 0 such that for S ) 0 the following holds: C1
d H Ž Fper Ž z, y 0 , S . , F Ž z, y 0 , S . . F
S
.
From w14, Proposition 3.2x it follows that there is another constant C2 G 0 such that for S ) 0 the following holds: d H Ž F Ž z . , F Ž z, y 0 , S . . F
C2
'S
.
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PERIODIC NEAR OPTIMAL CONTROL
Thus we have for all z g M, y 0 g N, and S G 1 max d H Ž Fper Ž z, y 0 , S . , F Ž z . . , d H Ž Fper Ž z, y 0 , S . , F Ž z, y 0 , S . .
½
F
C1 q C 2
'S
.
5 Ž 11 .
Žiii. By compactness of M, N, and ⍀ the vector fields f and f Ž1. are uniformly bounded. Thus there is a time T0 ) 0 small enough such that for any slow initial value z 0 g M we can deal with only one local parametrization, ⌽i : Ui = ⺢ m ª TM for B␣ Ž z 0 . ; M for all times t g w0, T0 x. If T0 - 1 we can rescale the vector fields f and g. So, w.l.o.g., we can assume that T0 s 1. In the following we will not change the notation for points and vectors on M in local coordinates. Živ. Let ⑀ ) 0. We consider the singularly perturbed control system Ž2. and the averaged slow subsystem Ž6. in the fast time [ tr⑀ , where accordingly g w0, 1r⑀ x. We divide this interval in subintervals of the form w l , lq1 x, which all have the same length S⑀ ) 0, except the last one, which may be shorter. Accordingly the index l is an element of the index set
½
I⑀ [ 0, . . . ,
1
⑀ S⑀
5
.
Žv. Let z⑀ g S⑀ Ž z 0 , y 0 . ; C Žw0, 1r⑀ x, ⺢ m . be a slow trajectory of the singularly perturbed control system Ž2. considered in this local coordinates. Then we have z⑀ Ž0. s z 0 and for l g I⑀ and g w l , lq1 x
z⑀ Ž . s z⑀ Ž l . q ⑀
H
f Ž z⑀ Ž s . , y⑀ Ž s . , u Ž s . . ds.
l
We define a family Ž z l . l g I⑀ ; ⺢ m by z 0 [ z 0 and lq1
z lq1 [ z l q ⑀
H
f Ž z l , y⑀ Ž s . , u Ž s . . ds
l
and an interpolating curve by zl Ž . [ zl q ⑀
H
f Ž z l , y⑀ Ž s . , u Ž s . . ds.
l
We define for g w l , lq1 x and l g I⑀ ⌬ l Ž . [ max
lFsF
z⑀ Ž s . y z l Ž s . .
142
G. GRAMMEL
Then we obtain
⌬l Ž . F ⌬l Žl . q ⑀
H
L Ž ⑀ S⑀ P q ⌬ l Ž s . . ds
l
with a Lipschitz constant L G 0 and an upper bound P G 0 for the slow vector field f on ⺢ m. We apply the Gronwall lemma and obtain for all l g I⑀ ⌬ lq1 Ž lq1 . F Ž ⌬ l Ž l . q ⑀ 2 S⑀2 LP . e ⑀ S⑀ L . We obtain by induction for all l g I⑀ ⌬ l Ž l . F ⑀ S⑀ LPe L . We choose a control value 1
lq1
H S ⑀
lŽ1.
g⍀
Ž1.
Ž 12 .
such that
f Ž z l , y⑀ Ž s . , u Ž s . . ds y f Ž1. Ž z l , lŽ1. . F
C1 q C 2
'S
⑀
l
and define a new family Žl . l g I⑀ ; ⺢ m by 0 [ z 0 and
lq1 [ l q ⑀ S⑀ f Ž1. Ž l , lŽ1. . . We define an interpolating curve by
l Ž . [ l q ⑀ Ž y l . f Ž1. Ž l , lŽ1. . . We can immediately estimate 5 z lq1 y lq1 5 F 5 z l y l 5 Ž 1 q L ⑀ S⑀ . q ⑀ S⑀
C1 q C 2
'S
⑀
from which follows 5 z l y l 5 F
C1 q C 2
'S
eL.
Ž 13 .
⑀
The control values lŽ1. g ⍀ Ž1. define a trajectory of the averaged control system by z 0 Ž0. s z 0 and z0 Ž . [ z0 Ž l . q ⑀
H
f Ž1. Ž z 0 Ž s . , lŽ1. . ds.
l
In order to estimate the distance of this trajectory to l Ž . we define for g w l , lq1 x and l g I⑀ D l Ž . [ max
lFsF
l Ž s . y z 0 Ž s . .
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PERIODIC NEAR OPTIMAL CONTROL
Then we obtain Dl Ž . F Dl Ž l . q ⑀
H
L Ž ⑀ S⑀ P q D l Ž s . . ds.
l
We apply the Gronwall lemma and get for all l g I⑀ D lq1 Ž lq1 . F Ž D l Ž l . q ⑀ 2 S⑀2 LP . e ⑀ S⑀ L . We obtain by induction for all l g I⑀ D l Ž l . F eS⑀ LPe L .
Ž 14 .
Finally we get, combining Ž12., Ž13., Ž14., for all g w0, 1r⑀ x z⑀ Ž . y z 0 Ž . F e L 2 ⑀ S⑀ LP q
ž
C1 q C 2
/
'S
⑀
F C3 ⑀ 1r3
Ž 15 .
for a constant C3 ) 0, if we define S⑀ [ ⑀y2 r3. Žvi. In the case T ) 1 we may not be able to manage with only one local parametrization of M. Nevertheless, we can use the following version of the Gronwall lemma for the averaged slow subsystem on M: There is a constant L0 G 0 such that for any z10 , z 20 g M, any control function uŽ1. g U Ž1., any time T G 1, and any t g w0, T x the following holds: d Ž z Ž t , z10 , uŽ1. . , z Ž t , z 20 , uŽ1. . . F e L 0 Td Ž z10 , z 20 . . The proof of this version of the Gronwall lemma relies on an application of the standard Gronwall lemma for systems in ⺢ m in local coordinates and on the estimate Ž10.. We proceed as follows: Let ⑀ ) 0 and T G 1. Let z⑀ g S⑀ Ž z 0 , y 0 . ; C Žw0, Tr⑀ x, M . be a slow trajectory of the singularly perturbed control system Ž2. on M. According to part Žv. of the proof there are trajectories z 0i Ž⭈. g S0 Ž z⑀ Ž i .. ; C Žw i, i q 1x, M . of the averaged slow subsystem Ž4. with z 0i Ž i . s z⑀ Ž i . ,
d Ž z⑀ Ž t . , z 0i Ž t . . F ␥ C3 ⑀ 1r3
for all i s 0, 1, 2, . . . and t g w i, i q 1x. Applying the version of the Gronwall lemma above we obtain a trajectory z 0 Ž⭈. of the averaged slow subsystem Ž4. with i
d Ž z⑀ Ž t . , z 0 Ž t . . F ␥ C3 ⑀ 1r3
Ý eL js0
0
j
144
G. GRAMMEL
for all i s 0, 1, 2, . . . w T x and t g w0, i q 1x. The required estimate follows easily with C [ ␥ C3 q L0 . Žvii. The proof of the reversal statement, namely the existence of a trajectory z⑀ Ž⭈. of the singularly perturbed control system Ž2. for a given trajectory z 0 Ž⭈. g S0 Ž z 0 . ; C Žw0, T x, M . of the averaged slow subsystem Ž4. with dŽ z⑀ Ž t ., z 0 Ž t .. F ⑀ 1r3 e CT for all t g w0, T x, is similar and we omit it. REFERENCES 1. Z. Artstein, Invariant measures of differential inclusions applied to singular perturbations, J. Differential Equations 152 Ž1999., 289᎐307. 2. F. Bagagiolo and M. Bardi, Singular perturbation of a finite horizon problem with state-space constraints, SIAM J. Control Optim. 36 Ž1998., 2040᎐2060. 3. R. L. Bishop and R. J. Crittenden, ‘‘Geometry of manifolds,’’ Academic Press, New YorkrLondon, 1964. 4. W. M. Boothby, ‘‘An Introduction to Differentiable Manifolds and Riemannian Geometry,’’ 2nd ed., Academic Press, Orlando, 1986. 5. F. Colonius and W. Kleimann, Infinite time optimal control and periodicity, Appl. Math. Optim. 20 Ž1989., 113᎐130. 6. F. Colonius and W. Kleimann, Limit behaviour and genericity for nonlinear control systems, J. Differential Equations 109 Ž1994., 8᎐41. 7. M. P. do Carmo, ‘‘Riemannian Geometry,’’ Birkhauser, Boston, 1992. ¨ 8. T. Donchev and I. Slavov, Averaging method for one-sided Lipschitz differential inclusions with generalized solutions, SIAM J. Control Optim. 37 Ž1999., 1600᎐1613. 9. H. S. Dumas and F. Golse, The averaging method for perturbations of mixing flows, Ergodic Theory Dynam. Syst. 17 Ž1997., 1339᎐1358. 10. V. G. Gaitsgory, Suboptimization of singularly perturbed control systems, SIAM J. Control Optim. 30 Ž1992., 1228᎐1249. 11. V. G. Gaitsgory, Suboptimal control of singularly perturbed systems and periodic control, IEEE Trans. Automatic Control 38 Ž1993., 888᎐903. 12. V. G. Gaitsgory, A new reduction technique for a class of singularly perturbed optimal control problems, IEEE Trans. Automat. Control 40 Ž1995., 721᎐724. 13. G. Grammel, Controllability of differential inclusions, J. Dynam. Control Systems 1 Ž1995., 581᎐595. 14. G. Grammel, Averaging of singularly perturbed systems, Nonlinear Anal. 28 Ž1997., 1851᎐1865. 15. G. Grammel, Estimate of periodic suboptimal controls, J. Optim. Theory Appl. 99 Ž1998., 681᎐689. 16. A. Isidori, ‘‘Nonlinear Control Systems,’’ 3rd ed., Springer-Verlag, Berlin, 1995. 17. W. Kliemann, Recurrence and invariant measures for degenerate diffusions, Ann. Probab. 15 Ž1986., 690᎐707. 18. P. V. Kokotovic, H. K. Khalil, and J. O’Reilly, ‘‘Singular Perturbation Methods in Control: Analysis and Design,’’ Academic Press, London, 1986. 19. E. Sontag, ‘‘Mathematical Control Theory,’’ 2nd ed., Springer-Verlag, New YorkrBerlin, 1998. 20. F. Wirth, Convergence of the value functions of discounted infinite horizon optimal control problems with low discount rates, Math. Oper. Res. 18 Ž1993., 1006᎐1013.
Periodic Near Optimal Control G. Grammel Center for Mathematics M6rH4, Technical Uni¨ ersity Munich, Arcisstr. 21, 80290 Munich, Germany E-mail: [email protected] Submitted by Robert E. O’Malley, Jr. Received December 28, 1998
We consider the infinite time optimal control problem of minimizing an average cost functional for nonlinear singularly perturbed systems. In particular, we are interested in the existence of near optimal trajectories which are periodic. For this purpose, we investigate nonlinear control systems with three time scales and construct a limiting system for the slowest motion by an iterated averaging procedure. We furthermore show that global tools, invariant control sets, and local tools, Lie algebra ranks of vector fields, are preserved under singular perturbations. The existence of periodic near optimal trajectories is verified under the condition that the fast subsystem and an unperturbed slow subsystem are controllable and that a standard Lie algebra rank condition for local accessibility holds. 䊚 2000 Academic Press
Key Words: nonlinear control system; periodic control; singular perturbation; near optimality; controllability; averaging.
1. INTRODUCTION We consider the dynamic optimization problem of minimizing an average cost functional 1 T l Ž z Ž t . , y Ž t . , u Ž t . . dt, Ž 1. Ž z 0 , y 0 , u . ¬ lim sup Tª⬁ T 0 where t ¬ z Ž t . s z Ž t, z 0 , y 0 , u. and t ¬ y Ž t . s y Ž t, z 0 , y 0 , u. are the trajectories of the singularly perturbed nonlinear control system
H
˙z Ž t . s f Ž z Ž t . , y Ž t . , u Ž t . . , ⑀˙ y Ž t . s g Ž y Ž t . , uŽ t . . ,
z Ž 0. s z 0 , y Ž 0. s y 0 ,
uŽ t . g ⍀ .
Ž 2.
Here, the singular perturbation parameter ⑀ ) 0 is assumed to be small in order to reflect that y Ž⭈. is much faster than z Ž⭈.. 124 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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A natural aspect arising with this problem is the property of possessing periodic optimal trajectories or at least periodic suboptimal trajectories. The minimization of an average cost functional is a typical infinite time problem, since, for any positive time T, only the costs produced on an infinite time interval w T, ⬁. are relevant in the limit, but not the costs produced on the finite interval w0, T x. On the other hand, existence of periodic Žsub-. optimal controls implies that the minimization problem basically can be reduced to a finite time horizon. By this reason, periodicity of Žsub-. optimal controls only can be expected under special structural properties of the nonlinear control system Ž2.. It is well known, and follows from results in w5x, that the optimization problem Ž1., Ž2. possesses suboptimal periodic trajectories and controls, if the system Ž2. is controllable. When the state space is compact and the system is locally accessible, controllability even is a necessary condition for the existence of periodic suboptimal controls for any average cost functional Žsee, e.g., w15x.. The controllability notion may be weakened, if the dynamics and the functional are somehow adapted. In w5x, it is shown that whenever suboptimal trajectories are contained in a region of complete controllability, a so-called control set, then there exist suboptimal periodic trajectories. Similar structural properties of the control system are required when approximating the average functional by discounted functionals; see, e.g., w20x. In connection with perturbations, the notion suboptimal becomes a little bit confusing. For perturbed systems one is usually interested in controls and trajectories which become closer and closer to optimal ones, as the perturbation tends to zero. Such controls are called near optimal. Moreover, the asymptotic behavior of the optimal value function, as the perturbation tends to zero, becomes an interesting topic, especially for optimization problems on the infinite time horizon. Notice that the averaging techniques for nonlinear control systems or for differential inclusions with two time scales, as developed in w1, 8, 10, 14x, merely result in approximation statements on a bounded time horizon. Theorem 2.5 contains the main result. If the fast subsystem and an averaged slow subsystem, both considered separately, are completely controllable and if a standard Lie algebra rank condition for local accessibility holds, then we have uniform convergence of the optimal value function to a constant function, as ⑀ ª 0, existence of periodic near optimal trajectories. 䢇
䢇
This is a typical singular perturbation result. Instead of assuming controllability of the whole system, we decompose the state space into an unperturbed slow and an unperturbed fast part and require controllability
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only of the decoupled unperturbed subsystems, which are both independent of the perturbation parameter. It is furthermore a nonlinear generalization of well-known results on linear systems. For linear systems, controllability of unperturbed slow and fast subsystems implies already the controllability of the whole singularly perturbed system, at least for small perturbation parameters; see w18x. The proof of Theorem 2.5 consists of two parts. First, we note that the problem has three natural time scales. For this reason we investigate nonlinear control systems with three time scales and construct a limiting system for the slowest motion. In contrast to common averaging approaches to the approximation of control systems with two time scales w10, 14x, we construct a limiting system for the slow motion as a control system and not as a differential inclusion. This method has the advantage that it can be iterated and thus allows an application to systems with multiple time scales. The second part of the proof is of a different nature. Since, for nonlinear singularly perturbed systems, controllability of the fast subsystem and of the averaged slow subsystem does not imply controllability of the whole singularly perturbed system, we furthermore use global tools from dynamical systems theory Žcontrol sets. and local tools from geometric control theory ŽLie algebras of vector fields. and investigate their asymptotic properties, as the singular perturbation parameter tends to zero. It turns out that both tools are somehow preserved under singular perturbations and allow one to obtain enough periodic orbits of the singularly perturbed system. The paper is organized as follows: In Section 2 we give the setting and recall some facts from differential geometry. We furthermore construct an averaged slow subsystem and state the main result. An example from stability theory closes the section. In Section 3 we apply the averaging technique to systems with three natural time scales. Section 4 contains stability statements for control sets and for the Lie algebra rank under singular perturbations. In Section 5 we fit together the results of the previous sections to prove the main result, Theorem 2.5. Section 6 contains the proof of the two scale averaging result introduced in Section 2.
2. PRELIMINARIES AND MAIN RESULT Regularity The state space of the singularly perturbed control system Ž2. is a product M = N, where M respectively N are compact C⬁-manifolds of dimension m respectively n. The control range is a compact metric space
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⍀ and the control functions u: w0, ⬁. ª ⍀ are measurable. The maps l: M = N = ⍀ ª ⺢, f : M = N = ⍀ ª TM, and g: N = ⍀ ª TN are continuous. For every fixed g ⍀ the vector field Ž l Ž⭈, ⭈ , ., f Ž⭈, ⭈ , ., g Ž⭈, .. is Lipschitz continuous on ⺢ = M = N. The regularity of the vector fields guarantees the unique existence of trajectories for any initial points and any control function; see, e.g., w19x. Notation. By d H we denote the Hausdorff metric. By U we denote the set of all measurable control functions u: w0, ⬁. ª ⍀, whereas elements of the control range ⍀ are denoted by . By A⑀ Ž t, Ž z 0 , y 0 .. [ Ž z Ž t, z 0 , y 0 , u., y Ž t, z 0 , y 0 , u.. g M = N: u g U 4 we denote the attainable set of the singularly perturbed system Ž2.. The A¨ eraging Method This order restriction method is based on the idea that, in the limit ⑀ ª 0, the fast state should not be considered explicitly anymore, but only its average influence on the slow dynamics should. This naturally requires some ergodic properties of the fast motion. In its natural time scale s tr⑀ the fast subsystem on N becomes
˙y Ž . s g Ž y Ž . , u Ž . . ,
y Ž 0. s y 0 ,
uŽ . g ⍀ .
Ž 3.
We denote its solutions by y Ž⭈, y 0 , u. and its attainable set by A N Ž , y 0 .. The following controllability assumption on the fast subsystem is essential. Assumption 2.1. The fast subsystem Ž3. is completely controllable on N: There is a finite time max G 0 such that for all y 0 g N the attainable set A N Žw0, max x, y 0 . coincides with N. This controllability assumption guarantees that the fast subsystem has ‘‘enough periodicity’’ and that in the limit all fast initial points have the same rights. It is much weaker than the controllability type condition introduced in w2x, where it is assumed that all sets of velocities g Ž y, . : g ⍀ 4 contain the balls B1Ž0. ; Ty N. In particular, we do not presume the existence of any equilibria of the fast flow on N. The construction of an averaged slow subsystem is similar to the construction introduced in w11, 12x, but without stability assumptions on the fast flow. As in w2, 11, 12x, in some sense, the fast motion will play the role of controls for the unperturbed slow system. For every point y 0 g N we define the subset U 0Ž y 0 . ; U , which consists of 0-periodic control functions u defining 0-periodic trajectories y Ž⭈, y 0 , u.. The subset U 0Ž y 0 . may be empty for certain periods 0 ) 0. But, by the controllability of the fast subsystem, we certainly have a nonvoid union D 0 ) 0 U 0Ž y 0 .. Any periodic fast trajectory of Ž3. defined by
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an initial state y 0 g N and by a corresponding periodic control function u g U 0Ž y 0 . generates an averaged vector field Ž1. on M:
Ž1. Ž z . s
1
0
0
H0 f Ž z, y Ž , y
0
, u . , u Ž . . d .
Let ⍀ Ž1. be the collection of all these vector fields together with its closure in C Ž M; TM .. Then ⍀ Ž1. can be considered as the control range of an averaged slow subsystem on M,
˙z Ž t . s f Ž1. Ž z Ž t . , uŽ1. Ž t . . ,
z Ž 0. s z 0 ,
uŽ1. Ž t . g ⍀ Ž1. ,
Ž 4.
where the right-hand side is defined by f Ž1. Ž z, Ž1. . [ Ž1. Ž z .. We denote its solutions by z 0 Ž⭈, z 0 , uŽ1. .. In order to get a richer controllability structure of the whole singularly perturbed system, we add an analogous controllability condition on the averaged slow subsystem Ž4.. We denote the attainable set for Ž4. by A M Ž t, z 0 .. Assumption 2.2. The averaged slow subsystem Ž4. is completely controllable on M: There is a finite time t max G 0 such that for all z 0 g M the attainable set A M Žw0, t max x, z 0 . coincides with M. Nevertheless, there are simple two-dimensional examples on the torus which show that this ‘‘composite controllability’’ as introduced with Assumptions 2.1 and 2.2 does not guarantee the existence of periodic trajectories of the singularly perturbed system. Thus, the standard Lie algebra rank condition on nonlinear control systems seems to be vital for the ‘‘synchronization’’ of the two subsystems. It requires higher regularity of the vector fields Ž f, g .. Assumption 2.3. For any fixed g ⍀ the vector field Ž f Ž⭈, ⭈ , ., g Ž⭈, .. is C⬁ on M = N. The Lie algebra generated by the vector fields f Ž⭈, ⭈, ., g Ž⭈, .. has full rank m q n at some Ž z 0 , y 0 . g M = N. DEFINITION 2.4. Consider the optimization problem Ž1., Ž2.. We define the corresponding optimal value function on M = N as V⑀ Ž z 0 , y 0 . [ inf
ug U
ž
lim sup Tª⬁
1
T lŽ zŽt, z H T 0
0
, y 0 , u Ž ⭈. . ,
y Ž t , z 0 , y 0 , u Ž ⭈ . . , u Ž t . . dt .
/
Now we are in a position to state the main result of the paper.
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THEOREM 2.5. ŽI. Suppose Assumptions 2.1 and 2.2 hold. Then there is a continuous function
: w 0, ⬁ . ª w 0, ⬁ .
with Ž 0 . s 0,
and a real number V0 g ⺢ such that for all Ž z 0 , y 0 . g M = N the estimate V⑀ Ž z 0 , y 0 . y V0 F Ž ⑀ . is ¨ alid. ŽII. If additionally Assumption 2.3 holds, then, for ⑀ ) 0 small enough, there exists a set I⑀ ; M = N with d H Ž I⑀ , M = N . F Ž ⑀ . , such that for all initial states Ž z 0 , y 0 . g I⑀ there exists a period t 0 ) 0 and a t 0-periodic control function u⑀ producing a t 0-periodic trajectory Ž z⑀ Ž⭈., y⑀ Ž⭈.. of the system Ž2. with 1 t0
t0
H0
l Ž z⑀ Ž t . , y⑀ Ž t . , u⑀ Ž t . . dt y V0 F Ž ⑀ . .
What follows is a short discussion of the conditions of this result. The controllability condition on the fast subsystem, Assumption 2.1, is very natural, if one wants to achieve that the averaged slow subsystem does not depend on fast initial states. We furthermore assume that the fast flow is decoupled, in the sense that the vector field g does not depend on the slow variable z. This restriction is necessary in order to avoid a significant loss of regularity of an averaged system. Moreover, our averaging procedure would not work this way, if we allow a fully coupled system, since it is not clear how the periodic trajectoriesrcontrols of the unperturbed fast flow depend on the slow state. The controllability condition on the averaged slow subsystem, Assumption 2.2, may be difficult to verify for practical problems. Nevertheless, it is a condition for an ⑀-independent system on a lower dimensional space M and for this reason it is easier to check than controllability of the full system on M = N for e¨ ery ⑀ ) 0. Assumption 2.3 is reduced to the minimum, since we assume full rank of the Lie algebra in only one point of the state space. We give a low-dimensional example from stability theory. EXAMPLE 2.6. We consider a singularly perturbed controlled oscillator defined by 0
˙s Ž t . s y1 q y Ž t . u Ž t . 1
1 s t , 0 Ž .
⑀˙ yŽ t. s
0 y1
1 yŽ t. , 0
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where sŽ t . s Ž s1Ž t ., s2 Ž t .. g ⺢ 2 and y Ž t . s Ž y 1Ž t ., y 2 Ž t .. g ⺢ 2 with 5 y Ž t .5 ' 1. Let the control range ⍀ be an interval w min , max x. We investigate the exponential stability of the slow subsystem via the exponential growth rates
⑀ Ž s 0 , y 0 , u . s lim sup Tª⬁
1 T
log Ž s Ž T , s 0 , y 0 , u .
..
The projection onto the sphere M s S 1 defines a slow subsystem for z Ž t . s sŽ t .r5 sŽ t .5: 0
1 z t 0 Ž .
˙z Ž t . s y1 q y Ž t . u Ž t . 1
0 y zŽ t. , y1 q y 1 Ž t . u Ž t .
¦
1 z t z t . 0 Ž . Ž .
;
Moreover, in these coordinates, the exponential growth rates can be written as an average functional
⑀ Ž z 0 , y 0 , u . s lim sup Tª⬁
1
T
H T 0
¦
zŽ t. ,
0 y1 q y 1 Ž t . u Ž t .
1 z t dt. 0 Ž .
;
It is obvious that the fast subsystem on N s S 1 is periodic. Hence, Assumption 2.1 is satisfied and we can construct an averaged slow subsystem. The averaged vector fields are of the form
Ž1. Ž z . s
1
2 k
H 2 k 0
ž
0 y1 q y 1 Ž t . u Ž t .
¦
y z,
1 z 0
0 y1 q y 1 Ž t . u Ž t .
1 z z dt, 0
;/
where k g ⺞ and u g U . In particular, for k s 1 and uŽ t . ' min , we obtain an averaged vector field
Ž1. Ž z . s
0 y1
1 0 z y z, 0 y1
¦
1 0 z zs 0 y1
;
1 z, 0
which produces periodic trajectories on S 1 for the averaged slow subsystem. Hence, Assumption 2.2 is satisfied and we can apply Theorem 2.5, part ŽI., and obtain that the minimal exponential growth rates converge, as ⑀ ª 0.
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As for Assumption 2.3, we see immediately that the vectors
and
0 y1 q y 1 min
1 0 0 z y z, y1 q y 1 min
¦
0 y1
0 y1 q y 1 max
1 y 0
1 0 0 z y z, y1 q y 1 max
¦
0 y1
1 y 0
1 0 z z
;
0 ; 0
1 0 z z
are linearly independent in Tz M = Ty N, if min - max and if z1 y 1 / 0. Thus, Assumption 2.3 is satisfied and we conclude that the minimal exponential growth rates can be Žnearly. realized by periodic controls.
3. ORDER REDUCTION VIA ITERATED AVERAGING The averaged slow subsystem serves us as a nominal system for the slow motion in the following sense: Let S⑀ Ž z 0 , y 0 . ; C Žw0, T x, M . be the set of all slow trajectories z Ž⭈. of the singularly perturbed control system Ž2. and let S0 Ž z 0 . ; C Žw0, T x, M . be the set of all trajectories of the averaged slow subsystem Ž4., both obtained for times t g w0, T x. Then we have: LEMMA 3.1. Suppose Assumption 2.1 holds. The control range ⍀ Ž1. of the a¨ eraged slow subsystem Ž4. is a compact metric space, the map f Ž1. : M = ⍀ Ž1. ª TM is continuous, and for e¨ ery fixed Ž1. g ⍀ Ž1. the ¨ ector field f Ž1. Ž⭈, Ž1. . is Lipschitz continuous on M. Furthermore there is a constant C G 0 such that for all initial ¨ alues Ž z 0 , y 0 . g M = N, perturbation parameters ⑀ ) 0, and times T G 1 the following estimate holds: d H Ž S⑀ Ž z 0 , y 0 . , S0 Ž z 0 . . F e C T⑀ 1r3 . The proof is contained in the last section. Lemma 3.1 is of interest in its own right for two reasons: It provides an averaging procedure which preserves the system’s structure and it gives explicit approximation rates of order O Ž ⑀ 1r3 . for the uniform convergence of the slow trajectories on finite time intervals. It is remarkable that the rate of convergence coincides with the one obtained in w9x. There, a singularly perturbed ODE with decoupled fast
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flow is considered and it is shown that the average rate of convergence of the slow trajectories is of order O Ž ⑀ 1r3 ., where the average is taken over the fast initial states in N with respect to a canonical measure. The construction of the averaged slow subsystem is different than the usual averaging techniques, since it preserves the regularity of the system and thus a re-iteration of the procedure becomes possible. In contrast, the usual averaging techniques, as in w10, 14x, produce an averaged slow subsystem in the form of a differential inclusion. Notice that the dynamic optimization problem Ž1., Ž2. possesses three natural time scales. Since ⑀ ) 0 is assumed to be small, y Ž⭈. moves faster than z Ž⭈.. On the other hand the integrand of the cost functional becomes smaller and smaller since we are interested in the situation T ª ⬁. For this reason we study the approximation properties of an augmented control system, which is set up by introducing a new trajectory cŽ⭈. and a perturbation parameter ␦ ) 0:
˙c Ž t . s ␦ e Ž c Ž t . , z Ž t . , y Ž t . , u Ž t . . ,
c Ž 0. s c 0 ,
˙z Ž t . s f Ž z Ž t . , y Ž t . , u Ž t . . ,
z Ž 0. s z 0 ,
⑀˙ y Ž t . s g Ž y Ž t . , uŽ t . . ,
y Ž 0. s y , 0
Ž 5. uŽ t . g ⍀ .
We assume that the state space of the additional cŽ⭈.-trajectory, K, is the Euclidean space ⺢ k or a compact C⬁-manifold of dimension k. The map e: K = M = N = ⍀ ª TK is continuous and for any fixed g ⍀ the map eŽ⭈, ⭈ , ⭈ , . is Lipschitz continuous on K = M = N. The connection with the optimization problem becomes clear by setting K s ⺢, eŽ c, z, y, . [ l Ž z, y, ., c 0 [ 0, and ␦ [ 1rT and by investigating the augmented system on the time interval t g w0, 1r␦ x. We work on this more general augmented system because all approximation statements that follow in this section are valid in this generality. The optimization problem suggests that one first considers the limit ␦ ª 0 for a fixed perturbation parameter ⑀ ) 0 and then investigates the behavior of suboptimal trajectories for this problem, as ⑀ ª 0. But from an approximation point of view it seems to be more natural to first eliminate the fastest motion y Ž⭈. and then to investigate the optimization problem for a limit system describing the situation ⑀ s 0. The latter procedure of eliminating the fastest motion furthermore has the advantage that it can be iterated. So a limiting system describing the limit Ž ⑀ , ␦ . s Ž0, 0. for the slowest motion cŽ⭈. canonically can be constructed. We start the order reduction procedure by applying Lemma 3.1 and considering Ž cŽ⭈., z Ž⭈.. as the slow and y Ž⭈. as the fast trajectories of the augmented system Ž5.. The construction of an averaged system for the Ž cŽ⭈., z Ž⭈.. trajectories representing the situation ⑀ s 0 is the same as
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before: Any periodic fast trajectory of Ž3. defined by an initial value y 0 g N and by a corresponding periodic control function u g U 0Ž y 0 . generates an averaged vector field Ž1. on K = M:
Ž1. Ž c, z . s
1
0
0
H0
ž
e Ž c, z, y Ž , y 0 , u . , u Ž . . f Ž z, y Ž , y 0 , u . , u Ž . .
/
d .
Let ⍀ Ž1. be the collection of all these vector fields together with its closure in C Ž K = M; TK = TM .. Then ⍀ Ž1. can be considered as the control range of an averaged control system on K = M:
˙c Ž t . s ␦ e Ž1. Ž c Ž t . , z Ž t . , uŽ1. Ž t . . ,
c Ž 0. s c 0 ,
˙z Ž t . s f Ž1. Ž z Ž t . , uŽ1. Ž t . . ,
z Ž 0. s z 0 ,
uŽ1. Ž t . g ⍀ Ž1. .
Ž 6.
We use the same notation f Ž1., uŽ1., ⍀ Ž1. for the augmented system Ž6. as for Ž4., since the control action on the Ždecoupled. system for z Ž⭈. is the same. By Assumption 2.2, for every point z 0 g M we can define the subset Ž1. Ž 0 . Ut 0 z ; U Ž1., which consists of t 0-periodic control functions uŽ1. defining t 0-periodic trajectories z Ž⭈, z 0 , uŽ1. . of Ž6.. Any t 0-periodic z Ž⭈, z 0 , uŽ1. . trajectory of Ž6. generates an averaged vector field Ž2. on K
Ž2. Ž c . s
1 t0
t 0 Ž1.
H0
e
Ž c, z Ž t , z 0 , uŽ1. . , uŽ1. Ž t . . dt.
Again, considering the closure of these vector field in C Ž K, TK . as a new control range ⍀ Ž2., we get a limit system for the c-motion on K:
˙c Ž t . s ␦ e Ž2. Ž c Ž t . , uŽ2. Ž t . . ,
c Ž 0. s c 0 ,
uŽ2. Ž t . g ⍀ Ž2. ,
Ž 7.
To describe the connection between this limit system Ž7. on K and the original augmented control system Ž5. on K = M = N, let SŽ ⑀ , ␦ .Ž c 0 , z 0 , y 0 . ; C Žw0, 1r␦ x; K ., respectively SŽ0, 0.Ž c 0 . ; C Žw0, 1r␦ x; K ., be the set of solutions of the augmented system Ž5., projected onto the slowest motion cŽ⭈., respectively of the limit system Ž7.. PROPOSITION 3.2. Suppose Assumptions 2.1 and 2.2 hold. The control range ⍀ Ž2. of the limit system Ž7. is a compact metric space, the map e Ž2. : K = ⍀ Ž2. ª TK is continuous, and for e¨ ery fixed Ž2. g ⍀ Ž2. the ¨ ector field e Ž2. Ž⭈, Ž2. . is Lipschitz continuous on K. Furthermore there are continuous functions
 , ␦ : w 0, ⬁ . ª w 0, ⬁ .
with  Ž 0 . s ␦ Ž 0 . s 0
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such that for all initial ¨ alues Ž c 0 , z 0 , y 0 . g K = M = N and perturbation parameters ⑀ ) 0 the following estimate holds: d H Ž SŽ ⑀ , ␦ Ž ⑀ .. Ž c 0 , z 0 , y 0 . , SŽ0 , 0. Ž c 0 . . F  Ž ⑀ . . Proof. The regularity of e Ž2. and the approximation statement followed by an iterated application of Lemma 3.1: Let SŽ0, ␦ .Ž c 0 , z 0 . ; C Žw0, 1r␦ x, K . be the set of solutions of Ž6., projected onto the slowest motion of cŽ⭈.. Lemma 3.1 applied to Ž5. and Ž6. gives the estimate d H Ž SŽ ⑀ , ␦ . Ž c 0 , z 0 , y 0 . , SŽ0 , ␦ . Ž c 0 , z 0 . . F e C r ␦⑀ 1r3 . Lemma 3.1 applies to Ž6. and Ž7. gives the estimate d H Ž SŽ0 , ␦ . Ž c 0 , z 0 . , SŽ0 , 0. Ž c 0 . . F e C␦ 1r3 . Thus we have for all Ž c 0 , z 0 , y 0 . g K = M = N d H Ž SŽ ⑀ , ␦ . Ž c 0 , z 0 , y 0 . , SŽ0 , 0. Ž c 0 . . F e C r ␦⑀ 1r3 q e C␦ 1r3 , and the claim follows, e.g., setting ␦ Ž ⑀ . [ yCrlogŽ ⑀ 1r6 . for ⑀ g Ž0, 1r2x.
4. CONTROL SETS AND LOCAL ACCESSIBILITY In this section we use the ‘‘composite controllability,’’ introduced with Assumptions 2.1 and 2.2, to get global approximate controllability properties of the singularly perturbed system Ž2. for sufficiently small perturbation parameters ⑀ ) 0. We furthermore show that the standard Lie algebra rank condition, Assumption 2.3, is robust under singular perturbations. Since the Lie algebraic condition is related to local controllability properties, we get a complete picture of the controllability structure of the singularly perturbed system Ž2.. In the sequel we study forward invariant subsets of the state space, in which one can steer at least approximately from any point to any other. These subsets are introduced in w17x, where they are called in¨ ariant control sets. For more details and perturbation results of control sets consult, e.g., w6, 13x. DEFINITION 4.1. We call a subset C ; M = N an invariant control set of Ž2. if clos C s clos AŽw0, ⬁., Ž z 0 , y 0 .. for all Ž z 0 , y 0 . g C and if C is maximal Žw.r.t. set inclusion. with this property. We show that the invariant control sets of the singularly perturbed system Ž2. are becoming denser in M = N, as the perturbation becomes smaller.
PERIODIC NEAR OPTIMAL CONTROL
135
LEMMA 4.2. Suppose Assumptions 2.1 and 2.2 hold. For all ⑀ ) 0 there is at least one closed in¨ ariant control set C⑀ ; M = N of the singularly perturbed control system Ž2.. There is a continuous function : w0, ⬁. ª w0, ⬁. with Ž0. s 0 such that for any family of in¨ ariant control sets C⑀ ; M = N 4⑀ ) 0 of the singularly perturbed control system Ž2. the following estimate holds: d H Ž C⑀ , M = N . F Ž ⑀ . . Proof. By compactness of M = N there exists at least one closed invariant control set C⑀ ; M = N for any ⑀ ) 0. This can be proved as follows: We assume that the right-hand side of the singularly perturbed system Ž2. is convex in TŽ z, y.Ž M = N . for all Ž z, y . g M = N. Then the attainable sets define a continuous semi-flow on the system of compact subsets of M = N. By Zorn’s lemma there is a compact subset C⑀ which is invariant under the attainable set semi-flow and minimal with this property; i.e., no other compact subset of C⑀ is invariant. Obviously this minimal invariant set C⑀ is a closed invariant control set of the singularly perturbed system. If the right-hand sides of the system are not convex, we still can work with the corresponding convex valued differential inclusion, and the obtained invariant control set also is a control set for the nonconvex system by well-known relaxation theorems. We take a family of closed invariant control sets C⑀ ; M = N 4⑀ ) 0 . Let Ž z⑀0 , y⑀0 . g C⑀ . By Lemma 3.1 and by Assumption 2.2 the projection ⌸ M A⑀ Žw0, 1 q t max x, Ž z⑀0 , y⑀0 .. onto M fulfills the estimate d H Ž ⌸ M A⑀ Ž w 0, 1 q tmax x , Ž z⑀0 , y⑀0 . . , M . F e CŽ1qt ma x .⑀ 1r3 . By Assumption 2.2 the projection ⌸ N A⑀ Žw0, max ⑀ x, Ž z⑀0 , y⑀0 .. onto N coincides with N, whereas obviously d H Ž ⌸ M A⑀ Ž w 0, max ⑀ x , Ž z⑀0 , y⑀0 . . , z⑀0 . F P˜⑀max , where P˜ G 0 is a uniform upper bound for 5 f 5. Since A⑀ Žw0, 1 q t max q max x, Ž z⑀0 , y⑀0 .. ; C⑀ we conclude that d H Ž C⑀ , M = N . F e C Ž1qt ma x .⑀ 1r3 q P˜⑀max , and the proof is finished. DEFINITION 4.3. The singularly perturbed system Ž2. is locally accessible in Ž z 0 , y 0 . g M = N, if for all t 0 ) 0 int A⑀ Ž w 0, t 0 x , Ž z 0 , y 0 . . / ⭋
and
int A⑀ Ž w yt 0 , 0 x , Ž z 0 , y 0 . . / ⭋.
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G. GRAMMEL
For nonlinear control systems, which are produced by C⬁-vector fields, local accessibility can be verified via the calculation of the Lie products and of the corresponding Lie algebra. If the Lie algebra L , generated by all admissible vector fields, has full rank, then the control system is locally accessible Žcompare w16, 19x.. Now we investigate the dependence of the Lie algebra rank condition for local accessibility on the singular perturbation parameter. Since the limit vector fields Ž0, g Ž⭈, .. s lim ⑀ ª 0 Ž ⑀ f Ž⭈, ⭈ , ., g Ž⭈, .. are degenerated on M = N, it is not obvious whether this condition is fulfilled for sufficiently small singular perturbation parameters. For more convenience we consider the vector fields in the natural scale of the fast motion. LEMMA 4.4. Suppose Assumption 2.3 holds. Then there is an ⑀ 0 ) 0 and a radius r ) 0 such that for all ⑀ g Ž0, ⑀ 0 x and all Ž z, y . g Br Ž z 0 , y 0 . [ Ž z, y . g M = N : dŽŽ z, y ., Ž z 0 , y 0 .. F r 4 the Lie algebra L⑀ , generated by the ¨ ector fields Ž ⑀ f Ž⭈, ⭈ , ., g Ž⭈, .., has full rank m q n at Ž z, y .. Proof. By continuity of the vector fields, there is a radius r ) 0 such that the Lie algebra L1 , generated by the vector fields Ž f Ž⭈, ⭈ , ., g Ž⭈, .., has full rank m q n at Ž z, y ., for all Ž z, y . g Br Ž z 0 , y 0 .. We fix a pair Ž z, y . g Br Ž z 0 , y 0 .. For ⑀ s 1 the linear space in TŽ z, y .Ž M = N . generated by the Lie algebra L⑀ coincides with the full tangent space TŽ z, y.Ž M = N .. In other words, for ⑀ s 1, there are m q n vectors ¨ 1Ž z, y, ⑀ ., . . . , ¨ mq nŽ z, y, ⑀ .4 in the Ž m q n.-dimensional tangent space TŽ z, y .Ž M = N ., which are defined by vector fields ¨ 1Ž⭈, ⭈ , ⑀ ., . . . , ¨ mqnŽ⭈, ⭈ , ⑀ .4 ; L⑀ on M = N such that det Ž ¨ 1 Ž z, y, ⑀ . , . . . , ¨ mqn Ž z, y, ⑀ . . / 0.
Ž 8.
We observe that the determinant detŽ ¨ 1Ž z, y, ⑀ ., . . . , ¨ mqnŽ z, y, ⑀ .. can be written as a real polynomial in ⑀ . We conclude that there is an ⑀Ž z, y . ) 0 such that Ž8. holds for all ⑀ g Ž0, ⑀Ž z, y . x. The claim follows by compactness of Br Ž z 0 , y 0 .. Lemmas 4.2 and 4.4 show that invariant control sets and Lie algebraic conditions on the vector fields are in some sense robust under singular perturbations. Together they ensure that invariant control sets have nonvoid inferior for sufficiently small perturbation parameters.
5. PROOF OF THEOREM 2.5 Theorem 2.5 consists of two parts. In the proof of the first part, ŽI., we only use Assumptions 2.1 and 2.2 to show via a three scale analysis Žsee Proposition 3.2 above. the existence of Žnot necessarily periodic. near optimal trajectories for any initial state Ž z 0 , y 0 . of Ž2..
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PERIODIC NEAR OPTIMAL CONTROL
In the proof of the second part, ŽII., we make use of the controllability type results obtained in the previous section in order to show that for sets of initial pairs Ž z 0 , y 0 ., which become denser in M = N, the near optimal trajectories can be closed to periodic near optimal trajectories. PROOF OF ŽI.. We first show that to every initial point Ž z 0 , y 0 . g M = N corresponds a family of near optimal, not necessarily periodic, control functions. Let Ž z 0 , y 0 . g M = N and ⑀ ) 0. We set T [ T⑀ [ 1r␦ Ž ⑀ ., where ␦ Ž ⑀ . ) 0 is given in Proposition 3.2. Then we get a three time scale control system Ž5. by setting e Ž c, z, y, . [ l Ž z, y, . ,
K [ ⺢,
␦[␦Ž⑀.,
c 0 s 0.
By Proposition 3.2 we can build up a limit system Ž7. for the slowest motion cŽ⭈.. Since the vector fields eŽ⭈, ⭈ , . do not depend on c g K, Ž7. equivalently can be written as
˙c Ž t . s uŽ2. Ž t . ,
c Ž 0 . s 0,
uŽ2. Ž t . g ⍀ Ž2. .
Now let T [ kT⑀ for a natural number k. Then for control functions u: w0, T x ª ⍀ the slowest motion of the augmented system Ž5. corresponding to the initial value c 0 [ 0 fulfills cŽ T . s s
1 T 1 k
T
H0
k
Ý is1
l Ž z Ž t . , y Ž t . , u Ž t . . dt 1
iT⑀
H l Ž z Ž t . , y Ž t . , u Ž t . . dt. T Ž iy1.T ⑀
⑀
According to Proposition 3.2 for all control functions u: wŽ i y 1.T⑀ , iT⑀ x ª ⍀ the following holds: 1
iT⑀
H l Ž z Ž t . , y Ž t . , u Ž t . . dt G min T Ž iy1.T ⑀
Ž2.
g ⍀ Ž2. 4 y  Ž ⑀ . .
⑀
Again by Proposition 3.2 the control function u: wŽ i y 1.T⑀ , iT⑀ x ª ⍀ can be chosen such that 1
iT⑀
H l Ž z Ž t . , y Ž t . , u Ž t . . dt F min T Ž iy1.T ⑀
Ž2.
g ⍀ Ž2. 4 q  Ž ⑀ . .
⑀
Then we obtain c Ž T . y min Ž2. g ⍀ Ž2. 4 F  Ž ⑀ . .
Ž 9.
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G. GRAMMEL
Since k is arbitrary, we conclude that for all Ž z 0 , y 0 . and ⑀ ) 0 the following holds: V⑀ Ž z 0 , y 0 . y min Ž2. g ⍀ Ž2. 4 F  Ž ⑀ . . Thus, every initial value Ž z 0 , y 0 . g M = N corresponds to a family of near optimal control functions. Furthermore the optimal values V⑀ Ž z 0 , y 0 . uniformly converge to V0 [ min Ž2. g ⍀ Ž2.4 . Proof of ŽII.. We now show that for initial values Ž z 0 , y 0 . g M = N, which lie in the interior of invariant control sets, the corresponding near optimal controls can be closed to periodic near optimal controls. We define I⑀ [ int C⑀ for invariant control sets C⑀ ; M = N of the singularly perturbed control system Ž2.. By Lemma 4.4 there is an ⑀ 0 ) 0, a radius r ) 0, and a point Ž z 0 , y 0 . g M = N such that the singularly perturbed control system is locally accessible in Br Ž z 0 , y 0 . for ⑀ g Ž0, ⑀ 0 x. On the other hand, for Ž ⑀ . - r we have C⑀ l Br Ž z 0 , y 0 . / ⭋. By the forward invariance of the invariant control set C⑀ we conclude I⑀ / ⭋ for ⑀ g Ž0, ⑀ 0 x and that furthermore C⑀ s clos I⑀ . By Lemma 4.2 we have d H Ž I⑀ , M = N . F Ž ⑀ .. We take Ž z 0 , y 0 . g I⑀ and ⑀ g Ž0, ⑀ 0 x. There is a time t⑀ Ž z 0 , y 0 . G 0 such that all points in C⑀ can be steered to Ž z 0 , y 0 . in a positive time smaller than t⑀ Ž z 0 , y 0 .. This follows easily from the local accessibility and the compactness of C⑀ : By local accessibility there is an open ball B ; I⑀ g Ž z, y . g M = N: Ž z 0 , y 0 . g A⑀ Žw0, 1x, Ž z, y ..4 and, by compactness of C⑀ , all points of C⑀ can be steered into this ball in uniformly bounded time. Thus any trajectory starting in Ž z 0 , y 0 . g I⑀ , after arbitrary long time, can be steered back to Ž z 0 , y 0 . g I⑀ in uniformly bounded time. We take a natural number k large enough that 2 ⍀ Ž2. t⑀ Ž z 0 , y 0 . kT⑀
F Ž⑀.
and set T [ kT⑀ . A periodic near optimal control function is constructed by setting u⑀ Ž t . [ u Ž t .
for t g w 0, T x ,
where u: w0, T x ª ⍀ is chosen according to Ž9., and by extending it on w T, T q t⑀ Ž z 0 , y 0 .x in such a way that for a time t 0 g w T, T q t⑀ Ž z 0 , y 0 .x the trajectory of the singularly perturbed control system Ž2. is back in Ž z 0 , y 0 . g I⑀ ; C⑀ . Then a straightforward calculation shows that 1 t0
t0
H0
l Ž z Ž t . , y Ž t . , u⑀ Ž t . . dt F V⑀ Ž z 0 , y 0 . q 2  Ž ⑀ . .
Finally, we define ⑀ [ max 3  Ž ⑀ ., ⑀ 4 and the proof is finished.
PERIODIC NEAR OPTIMAL CONTROL
139
6. PROOF OF LEMMA 3.1 We recall some facts from differential geometry which will be needed in the following proof. We assume that the manifold M is provided with a fixed Riemannian metric which naturally defines a metric dŽ⭈, ⭈ . on the tangent bundle TM Žcompare w7, pp. 78᎐80x.. Since M is compact we can manage with a finite family of local parametrizations i : ⺢ m > Ui ª M to cover the whole manifold. Note that a local parametrization i : ⺢ m > Ui ª M of the manifold M automatically extends to a local parametrization ⌽i : Ui = ⺢ m ª TM of the tangent bundle TM Žcompare w4, pp. 335, 336x.. Furthermore, again by compactness of M, there is a positive number ␣ ) 0 such that for all z g M there is a local parametrization whose image on M contains the closed ball B␣ Ž z . ; M. Finally we remark that the distances on the tangent bundle locally can be estimated by distances in Ui = ⺢ m and vice versa Žsee w3, pp. 126, 126x.. To be more precise: There is a constant ␥ ) 0 such that for all z g M, all z k g B␣ Ž z . ; M, and all ¨ k g Tz k M Ž k s 1, 2. the following holds: y1 ␥ d Ž Ž z1 , ¨ 1 . , Ž z1 , ¨ 2 . . F ⌽y1 i Ž z1 , ¨ 1 . y ⌽i Ž z 2 , ¨ 2 .
F
1
␥
d Ž Ž z1 , ¨ 1 . , Ž z1 , ¨ 2 . . .
Ž 10 .
By L G 0 we denote the Lipschitz constant of the functions f Ž⭈, y, ., considered in local coordinates. According ␥ 2 L G 0 is a Lipschitz constant of the functions f Ž⭈, y, ., considered on the manifold M. By P G 0 we denote the upper bound of the functions 5 f Ž⭈, y, .5, considered in local coordinates. Accordingly ␥ P G 0 is an upper bound of the functions dŽŽ⭈, 0., Ž⭈, f Ž⭈, y, ...: M ª TM. In the proofs we do not change notations for points and tangent vectors when they are considered in local coordinates. We also assume that the manifold N is provided with a fixed Riemmanian metric. Proof of Lemma 3.1. Ži. Obviously, the averaged vector fields Ž1. on M, generated by periodic controls u g U 0Ž y 0 ., are globally Lipschitz continuous on M, since we have in local coordinates for z1 , z 2 g y1 B␣ Ž z 0 .
Ž1. Ž z1 . y Ž1. Ž z 2 . F
1
0
0
H0
f Ž z1 , y Ž , y 0 , u . , u Ž . . y f Ž z 2 , y Ž , y 0 , u . , u Ž . . d
F L 5 z1 y z 2 5
140
G. GRAMMEL
are accordingly with respect to the Riemannian metric d Ž Ž1. Ž z1 . , Ž1. Ž z 2 . . F ␥ 2 Ld Ž z1 , z 2 . . Thus, by the Arzela Ascoli theorem, the closure of these vector fields in C Ž M, TM . is compact and f Ž1. of the required regularity. Žii. The right-hand sides of the averaged slow subsystem Ž4.,
D f Ž1. Ž z, Ž1. . 4 ,
FŽ z. [
g⍀ Ž1. Ž1.
can be represented as limits of periodic a¨ erages Fper Ž z, y 0 , S . [
D
D
0g Ž0, S . ug U 0 Ž y 0 .
½
1
0
0
H0 f Ž z, y Ž , y
0
5
, u . , u Ž . . dt .
By construction of the averaged slow subsystem we have F Ž z . s clos D Fper Ž z, y 0 , S . . S)0
Note that by controllability of the fast subsystem, the right-hand sides F Ž z . do not depend on the initial value y 0 g N and furthermore necessarily are convex, since with increasing S G S0 ) 0 more and more convex combinations of Fper Ž z, y 0 , S0 . are included in Fper Ž z, y 0 , S .. In order to compare the right-hand sides F Ž z . and the periodic averages Fper Ž z, y 0 , S ., we compare both of them with the finite time a¨ erages F Ž z, y 0 , S . [ clos
D
ug U
½
1
S f Ž z, y Ž , y H S 0
0
5
, u . , u Ž . . d .
By controllability of the fast subsystem there is a constant C1 G 0 such that for S ) 0 the following holds: C1
d H Ž Fper Ž z, y 0 , S . , F Ž z, y 0 , S . . F
S
.
From w14, Proposition 3.2x it follows that there is another constant C2 G 0 such that for S ) 0 the following holds: d H Ž F Ž z . , F Ž z, y 0 , S . . F
C2
'S
.
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PERIODIC NEAR OPTIMAL CONTROL
Thus we have for all z g M, y 0 g N, and S G 1 max d H Ž Fper Ž z, y 0 , S . , F Ž z . . , d H Ž Fper Ž z, y 0 , S . , F Ž z, y 0 , S . .
½
F
C1 q C 2
'S
.
5 Ž 11 .
Žiii. By compactness of M, N, and ⍀ the vector fields f and f Ž1. are uniformly bounded. Thus there is a time T0 ) 0 small enough such that for any slow initial value z 0 g M we can deal with only one local parametrization, ⌽i : Ui = ⺢ m ª TM for B␣ Ž z 0 . ; M for all times t g w0, T0 x. If T0 - 1 we can rescale the vector fields f and g. So, w.l.o.g., we can assume that T0 s 1. In the following we will not change the notation for points and vectors on M in local coordinates. Živ. Let ⑀ ) 0. We consider the singularly perturbed control system Ž2. and the averaged slow subsystem Ž6. in the fast time [ tr⑀ , where accordingly g w0, 1r⑀ x. We divide this interval in subintervals of the form w l , lq1 x, which all have the same length S⑀ ) 0, except the last one, which may be shorter. Accordingly the index l is an element of the index set
½
I⑀ [ 0, . . . ,
1
⑀ S⑀
5
.
Žv. Let z⑀ g S⑀ Ž z 0 , y 0 . ; C Žw0, 1r⑀ x, ⺢ m . be a slow trajectory of the singularly perturbed control system Ž2. considered in this local coordinates. Then we have z⑀ Ž0. s z 0 and for l g I⑀ and g w l , lq1 x
z⑀ Ž . s z⑀ Ž l . q ⑀
H
f Ž z⑀ Ž s . , y⑀ Ž s . , u Ž s . . ds.
l
We define a family Ž z l . l g I⑀ ; ⺢ m by z 0 [ z 0 and lq1
z lq1 [ z l q ⑀
H
f Ž z l , y⑀ Ž s . , u Ž s . . ds
l
and an interpolating curve by zl Ž . [ zl q ⑀
H
f Ž z l , y⑀ Ž s . , u Ž s . . ds.
l
We define for g w l , lq1 x and l g I⑀ ⌬ l Ž . [ max
lFsF
z⑀ Ž s . y z l Ž s . .
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G. GRAMMEL
Then we obtain
⌬l Ž . F ⌬l Žl . q ⑀
H
L Ž ⑀ S⑀ P q ⌬ l Ž s . . ds
l
with a Lipschitz constant L G 0 and an upper bound P G 0 for the slow vector field f on ⺢ m. We apply the Gronwall lemma and obtain for all l g I⑀ ⌬ lq1 Ž lq1 . F Ž ⌬ l Ž l . q ⑀ 2 S⑀2 LP . e ⑀ S⑀ L . We obtain by induction for all l g I⑀ ⌬ l Ž l . F ⑀ S⑀ LPe L . We choose a control value 1
lq1
H S ⑀
lŽ1.
g⍀
Ž1.
Ž 12 .
such that
f Ž z l , y⑀ Ž s . , u Ž s . . ds y f Ž1. Ž z l , lŽ1. . F
C1 q C 2
'S
⑀
l
and define a new family Žl . l g I⑀ ; ⺢ m by 0 [ z 0 and
lq1 [ l q ⑀ S⑀ f Ž1. Ž l , lŽ1. . . We define an interpolating curve by
l Ž . [ l q ⑀ Ž y l . f Ž1. Ž l , lŽ1. . . We can immediately estimate 5 z lq1 y lq1 5 F 5 z l y l 5 Ž 1 q L ⑀ S⑀ . q ⑀ S⑀
C1 q C 2
'S
⑀
from which follows 5 z l y l 5 F
C1 q C 2
'S
eL.
Ž 13 .
⑀
The control values lŽ1. g ⍀ Ž1. define a trajectory of the averaged control system by z 0 Ž0. s z 0 and z0 Ž . [ z0 Ž l . q ⑀
H
f Ž1. Ž z 0 Ž s . , lŽ1. . ds.
l
In order to estimate the distance of this trajectory to l Ž . we define for g w l , lq1 x and l g I⑀ D l Ž . [ max
lFsF
l Ž s . y z 0 Ž s . .
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PERIODIC NEAR OPTIMAL CONTROL
Then we obtain Dl Ž . F Dl Ž l . q ⑀
H
L Ž ⑀ S⑀ P q D l Ž s . . ds.
l
We apply the Gronwall lemma and get for all l g I⑀ D lq1 Ž lq1 . F Ž D l Ž l . q ⑀ 2 S⑀2 LP . e ⑀ S⑀ L . We obtain by induction for all l g I⑀ D l Ž l . F eS⑀ LPe L .
Ž 14 .
Finally we get, combining Ž12., Ž13., Ž14., for all g w0, 1r⑀ x z⑀ Ž . y z 0 Ž . F e L 2 ⑀ S⑀ LP q
ž
C1 q C 2
/
'S
⑀
F C3 ⑀ 1r3
Ž 15 .
for a constant C3 ) 0, if we define S⑀ [ ⑀y2 r3. Žvi. In the case T ) 1 we may not be able to manage with only one local parametrization of M. Nevertheless, we can use the following version of the Gronwall lemma for the averaged slow subsystem on M: There is a constant L0 G 0 such that for any z10 , z 20 g M, any control function uŽ1. g U Ž1., any time T G 1, and any t g w0, T x the following holds: d Ž z Ž t , z10 , uŽ1. . , z Ž t , z 20 , uŽ1. . . F e L 0 Td Ž z10 , z 20 . . The proof of this version of the Gronwall lemma relies on an application of the standard Gronwall lemma for systems in ⺢ m in local coordinates and on the estimate Ž10.. We proceed as follows: Let ⑀ ) 0 and T G 1. Let z⑀ g S⑀ Ž z 0 , y 0 . ; C Žw0, Tr⑀ x, M . be a slow trajectory of the singularly perturbed control system Ž2. on M. According to part Žv. of the proof there are trajectories z 0i Ž⭈. g S0 Ž z⑀ Ž i .. ; C Žw i, i q 1x, M . of the averaged slow subsystem Ž4. with z 0i Ž i . s z⑀ Ž i . ,
d Ž z⑀ Ž t . , z 0i Ž t . . F ␥ C3 ⑀ 1r3
for all i s 0, 1, 2, . . . and t g w i, i q 1x. Applying the version of the Gronwall lemma above we obtain a trajectory z 0 Ž⭈. of the averaged slow subsystem Ž4. with i
d Ž z⑀ Ž t . , z 0 Ž t . . F ␥ C3 ⑀ 1r3
Ý eL js0
0
j
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